# Tag Info

8

We assume that the process $\{r(t), \, t \ge 0\}$ satisfies an SDE of the form \begin{align*} dr(t) = \big( \theta(t) - a(t) r(t) \big)dt + \sigma(t) dW_t, \quad t > 0, \end{align*} where $\{W_t, \, t \ge 0 \}$ is a standard Brwonian motion. Note that \begin{align*} d\left(e^{\int_0^t a(u) du}r(t) \right) &=a(t) e^{\int_0^t a(u) du}r(t) dt + e^{\int_0^...

7

Hints: You know the vega of a digital call option formula: $V=-\frac{e^{-r(T-t)}}{\sigma} d_1 n\left(d_2\right)$ Where n is the standard normal density, which is positive. Sigma and exponential are also positive, so the sign of V is down to the sign of $d_1$. Which is negative when: $d_1 <0$ $\ln \frac{S}{X}+\left(r+0.5\sigma^2\right)(T-t)<0$ $S&... 7 At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with assuming arbitrage is not possible (otherwise it would be trivial wouldn't it). But, in my opinion, what you actually seek is the Efficient Markets Hypothesis.... 5 This is not quite true, in either direction. If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider a spot dynamics where the spot is a martingale that jumps up or down by integer amounts. The spot distribution is discrete, with zero density in between ... 5 Suppose that the given condition is true. You want to construct an arbitrage portfolio to take advantage of this. Now,$d$is an interest rate, and the condition suggests that$d$is too high. So you will want to receive$d$in order to profit. If you could, you would borrow money at$r$and lend it to the stock broker or exchange to collect the interest ... 5 What you need to do is to first make a variable change such as$u = \frac{x-\xi}{2\sqrt{t}}$. Then change the order of the limit and the integral. 4 The paper could be clearer indeed. It is a slightly confusing topic, but the important step here is to understand the consequence of the derivative$C$in the portfolio being priced at the assumed vol$\sigma$. This implies (by Black-Scholes) that it will by definition be true that:$\theta_t + \frac{\partial{C}}{\partial{S}}rS_t+ \frac{1}{2}\frac{\partial^...

4

The whole point of no-arbitrage pricing in a complete market is that a general underlying model of the form $$d S_t = \mu(S_t,t)\, dt + \sigma(S_t,t) \, dW_t$$ can be replaced with the risk-neutral process. $$d S_t = (r - \sigma^2/2)\, dt + \sigma(S_t,t) \, dW_t$$ for the purpose of finding the theoretical fair option price. This, of course, follows ...

4

Consider an (arithmetic) Ornstein-Uhlenbeck process as a model of the asset price $X_t$: $$dX_t = \kappa(\mu-S_t)dt + \sigma dW_t$$ where $\mu$ is the mean-reversion level, $\sigma$ is a volatility parameter, $W_t$ is Brownian motion, and $\kappa$ is the reversion speed. An Ornstein-Uhlenbeck process will revert to the mean infinitely often if \kappa >... 3 A very good book covering such fundamentals with no or only a minimal amount of maths — highly recommended! Puzzles of Finance: Six Practical Problems and Their Remarkable Solutions by Mark P. Kritzman The topics that are covered here are: Siegel's Paradox Likelihood of Loss Time Diversification Why the Expected Return Is Not To Be Expected Half Stocks ... 3 Write out the simple equations \begin{align} Y_j &= a_0 Z_j + a_1 Z_{j-1} + a_2 Z_{j-2}\\ Y_{j-1} &= a_0 Z_{j-1} + a_1 Z_{j-2} + a_2 Z_{j-3} \end{align} There are some very simple cases that makeY_j \perp Y_{j-1}$due to the independence assumption of the random variables$\{Z_i\}_{i\in\mathbb{Z}}$. An example is$a_0 \in \mathbb{R}\setminus \...

3

overall gamma is second derivative of whole portfolio over underlying. adding any function (such as underlying*constant) which second derivative is 0 does not alter overall second derivative.

2

Why do you think such a theorem would exist? I will give you a counterexample: You have two assets A and B. Both are completely identical in every respect, except price: The price of A is USD 1 the price of B is USD 2. Your strategy is simple: You (short) sell B for a gain of 2 and buy A for 1. This strategy requires no capital and leaves you with an ...

2

Let $X$ be endowed with the following partial order: $y \geq x$ means that $\Bbb P(y\geq x) = 1$. The AOA condition in your case states that the pricing law $p$ is strictly inctreasing with respect to $\geq$, whereas LOP says that $p$ is a linear function. Neither if the two implies another one in general. For example, if $X = \Bbb R$ then $p(x) = x^3$ is a ...

2

This doesn't really suffice as an existence proof, but you can start with a series of mathematical results collectively known as no free lunch theorems. The linked paper proves the average performance of any optimization algorithm over arbitrary problem domains is independent of the algorithm. That is, no single algorithm can ever be better than others on ...

2

This is where one needs the concept of no free lunch with vanishing risk (NFLVR), whose proof you can find in: Delbaen & Schachermayer (1994). Though, as a warning, I should mention it is pretty involved.

2

The comments above re all the entries of $\mu$ not being the same is true, but can be removed if you make the 2x2 determinant in question $\ge 0$ instead of $> 0$. The commenters know this of course. The answer to your question can be obtained by an application of the Cauchy-Schwartz inequality along with knowledge that a symmetric positive definite ...

2

In a local volatility model, which is inhomogeneous in space, you'll end up with having that implied volatility of a vanilla option $(K,T)$ is a function of the spot price $S$, i.e. $$\Sigma = \sigma(T,K,S)$$ As such when you compute the (total) derivative of the option price with respect to the spot price you'll have: \begin{align} \frac{d V}{d S} &= \...

2

If you can prove that $\kappa_{\alpha}(X,t)=a_X(\alpha,t^{-1})=t\ln(\frac{E[e^{t^{-1}X}]}{\alpha})$ is convex and apply the property of positive homogeneity, then the sub-additivity follows. In original paper, authors show that $\kappa_{\alpha}(X,t)=a_X(\alpha,t^{-1})$ is convex in $(X,t)$. Lemma: For fixed $\alpha$, all $\lambda\in[0,1],X,Y\in L_{M^+}$ ...

2

$\newcommand{\E}{\mathbb{E}}$ $\newcommand{\V}{\mathbb{V}}$ First note that \begin{eqnarray} \E[(r_{t + n} + 1)^2] &=& \E[r_{t+n}^2 + 2r_{t + n} + 1] = \E[r_{t + n}^2] + 2\mu + 1 \\&=& (\V[r_{t+n}] + \E^2[r_{t+n}]) + 2 \mu + 1 \\ &=& \sigma^2 + (\mu^2 + 2\mu + 1) = \sigma^2 + (\mu + 1)^2 \tag{1} \end{eqnarray} Now, since $r_{t+n}$...

2

Let $p\in(0,1)$. The corresponding quantile function of $X\sim N(\mu,\sigma^2)$ is given by $$F_X^{-1}(p)=\mu+\sigma\Phi^{-1}(p)=\mu+\sqrt{2}\sigma\mathrm{erf}^{-1}(2p-1),$$ where $\Phi^{-1}$ is the inverse of the cumulative distribution function of a standard normally distributed random variable and $\mathrm{erf}^{-1}$ is the inverted error function. Thus, ...

2

$E(X|X\geq b)=\frac{\int_b^{\infty}X dP}{P(X\geq b)}=\frac{\int_b^{a}X dP+\int_a^{\infty}X dP}{P(X\geq b)} \leq \frac{a\int_b^{a} dP+\int_a^{\infty}X dP}{P(X\geq b)}=\frac{a\int_b^{a} dP+\int_a^{\infty}X dP}{\int_b^{a} dP + P(X\geq a)}$ Now since $a \leq \frac{\int_a^{\infty}X dP}{P(X\geq a)}$, the right hand side of the equation above is smaller than or ...

1

Recall that for any deterministic function $g,$ Ito's integral follows a normal distribution: $$\int_0^t g(u) dW_u \sim N\left(0,\int_0^t g^2(u) du\right).$$ Therefore, since $$X_{t+s} - X_t = \int_t^{t+s} \frac{1}{\sqrt{1 + f(u)^2}} dB_u + \int_t^{t+s} \frac{f(u)}{\sqrt{1 + f(u)^2}} dW_u,$$ each integral follows a normal distribution and they are ...

1

This is explained in Hull. Alternatively you can check this link https://web.ma.utexas.edu/users/mcudina/m339d-lecture-ten-forwards-pricing.pdf Essentially the seller of the forward contract earns the income associated to the stock lending activity so it needs to be discounted from the forward price to ensure absence of arbitrage opportunity. Absence of ...

1

Let's go for a detailed and rigorous proof. Let us define our local currency $Y$ as the numéraire, i.e. the asset in terms of whose price the relative prices of all other tradeables are expressed. $X$ is therefore the foreign currency, whose price in terms of $Y$ is $X(t)$ at any time $t$. Let $r_X(t,T)$ be the risk-free interest rate in currency $X$ and $... 1 Let lnA be N(0,1) and lnB be N(0,k) where we will let k tend to zero. Then B has all of its density at 1, so A+B>1 in the limit. Hence A+B is not lognormal. 1 You're correct - the delta in terms of futures contracts is discount factor * N(d1) for a European call-on-future. Consider a zero-strike call as an example, the delta would be 1.0 multiplied by the discount factor. Maybe the author meant to say the delta in terms of forward contracts is N(d1)? 1 If I have read the question correctly then I will assume that$a$,$b$,$c$,$d$,$T_i$, and$k_i$are constants. If this is the case then the only term which we need to show is bounded is $$\big(a + b(T_i - t)\big)\exp\big(-c(T_i-t)\big).$$ If we assume that we are only considering the temporal domain$0 \leq t \leq T_i$such that$T_i - t \geq 0 $then ... 1 You're half way there. When$ R < D \ (< U) $, the return of the stock dominates the risk-free return in all states of the world. To benefit from that, just borrow cash and invest in the stock. At$t=0$this requires no net investment: borrowing cash means your account is credited$S_0$, while subsequently buying the stock suggests it is debited$S_0\$...

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